Question

9. Prove or disprove. a) If two undirected graphs have the same number of vertices, the same number of edges, the same number of cycles of each length and the same chromatic number, THEN they are isomorphic! b) A relation $R$ on a set $A$ is transitive iff $R^2 \subseteq R$. c) If a relation $R$ on a set $A$ is symmetric, then so is $R^2$. d) If $R$ is an equivalence relation and $[a]_R \cap [b]_R \neq \emptyset$, then $[a]_R = [b]_R$.

          9. Prove or disprove.
a) If two undirected graphs have the same number of vertices, the same number of
edges, the same number of cycles of each length and the same chromatic number,
THEN they are isomorphic!
b) A relation $R$ on a set $A$ is transitive iff $R^2 \subseteq R$.
c) If a relation $R$ on a set $A$ is symmetric, then so is $R^2$.
d) If $R$ is an equivalence relation and $[a]_R \cap [b]_R \neq \emptyset$, then $[a]_R = [b]_R$.

9. Prove or disprove.
a) If two undirected graphs have the same number of vertices, the same number of
edges, the same number of cycles of each length and the same chromatic number,
THEN they are isomorphic!
b) A relation R on a set A is transitive iff R^2 ⊆ R.
c) If a relation R on a set A is symmetric, then so is R^2.
d) If R is an equivalence relation and [a]R ∩ [b]R ≠∅, then [a]R = [b]R.

Added by Dennis Y.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Suman Saurav Thakur

Step 1

Two graphs can have the same number of vertices, edges, cycles of each length, and chromatic number, but still not be isomorphic. Isomorphism requires that the two graphs have the same structure, meaning that the vertices are connected in the same way. Just having Show more…

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a) If two undirected graphs have the same number of vertices, the same number of edges, the same number of cycles of each length, and the same chromatic number, THEN they are isomorphic! b) A relation R on a set A is transitive if and only if R^2 âŠ† R. c) If a relation R on a set A is symmetric, then R^2 is also symmetric. d) If R is an equivalence relation and [a]R â‰ [b]R, then [a]R = [b]R.